WELCOME

TQ SUMMER 2011 WORKSHOP:

PROPORTIONALITY

JUNE 6-17, 2011

GROUP NORMS

Be an active learner

Be an attentive listener

Be a reflective participant

Be conscious of your needs

and needs of others

Success in Algebra

What mathematics do students

need to know to be successful in

Algebra I?

Pre Test

Work on test items individually.Write your participant number on the test.

Our Focal Point for Proportional Reasoning

Students will be able to identify, analyze, and represent proportional

and non-proportional linear relationships, to transition

from proportional models a/b = c/d

to

direct variation models y = kx.

Understanding Proportionality

What do students need to know and be able to do to understand proportionality?

Chart ideas and be prepared to share your ideas with the whole group

Texas Response to Curriculum Focal Points for

Grade 8 MathematicsRepresenting, applying, and analyzing

proportionality

Students extend their understanding of proportionality to include representations on a coordinate plane and applications, including proportional changes.

TEA 2009

Fish is Fish

Let’s read a story by Leo Lionni.

Read the story individually.

Fish is Fish

What 2 or 3 messages does this story convey to you?

Discuss these at your table. Select someone who will share at least

one message from your table with the whole group.

PRINCIPLES OF LEARNING

Principle One:

Building on Prior Knowledge

Principle Two:

Understanding requires factual knowledge and conceptual framework

Principle Three: Metacognition

Principle One

Students come to the classroom with preconceptions about how the world works. If their initial understanding is not engaged, they may fail to grasp the new concepts and information that are taught or they may learn them for purposes of a test but revert to their preconceptions outside the classroom.

Principle Two

Achieving goals in mathematical understanding requires teachers to develop students’ knowledge networks, address students’ learning paths, and use multiple methods.

This suggests the importance of conceptual understanding, procedural fluency, and an effective organization of knowledge.

Principle Three

Metacognition, getting students to think about their thinking, can help students learn to take control of their own learning by defining learning goals and monitoring their progress in achieving them.

Atlantis 2008 Unit on Proportional

Reasoning

What is a unit?

Structure of the unit

Planning and Development of the Unit

Unit Framework

Pre-Assessmen

t

Post-Assessmen

t

Unit Map/

Learning Pathway

Unit Project/ Introductory

Activity

Unit Objectives

Language and Communication

Prior Knowledge/

Pre-Lesson

Generalization

Debugging

Application

Extension

Connection

Unit Core

Introductory Elements

Applied Elements

Core Elements

Outcomes

Identify and incorporate the three Principles of Learning

Understand proportional reasoning and connect to linear relationships

Transition from a single lesson lens to a unit learning lens

Experience the unit through teacher and student lenses

Unit Pre-Lesson

Student Hat: Complete the activity individually

Write your answers on the extra copy

Consider the following as you work:

What content is new and what is familiar to you as a student?

Unit Pre-Lesson

Teacher Hat: Table group discussion

What preconceptions might your students have? (misconceptions, good knowledge, and missing knowledge)

How does the pre-lesson connect with the three principles?

How will the pre-lesson inform classroom instruction?

Pre-Lesson: Teacher Notes

Review the teacher notes.

Highlight areas you want to remember.

What else would you add to the teacher notes?

Looking at Student Work

Analyze the student work on the

Pre-LessonWhat misconceptions did you find?How would you address these

misconceptions?

Another ProblemI went to the store and bought the same number of books as records. Books cost two dollars each and

records cost six dollars each. I spent $40 altogether.

Assuming that the equation 2B+6R=40 is correct, what is wrong if anything, with the following student

reasoning?

“2B+6R=40, since B=R, I can write 2B+6B=40, then 8B=40. This last equation says 8 books is equal to

$40. So, one book costs $5.”

REFLECTIONS

In your memo book, list two or three advantages to having a pre-lesson.

How does this connect to principle one?

(refer to summary pages in first tab of notebook)

Time for Reading

“From Words to Algebra: Mending Misconceptions ”

by Jack Lochhead and Jose P. Mestre

(Readings are in your binder under the Resources tab.)

Mark your Reading

! Insight

? Questions

√ Affirmation

From Words to Algebra: Mending Misconceptions

Table Group Discussion:

How does reading the article assist you in addressing misconceptions you identified in the student work?

From Words to Algebra: Mending Misconceptions

Table Group Discussion:

How will this impact your teaching?

BIG MATHEMATICAL IDEA

Goal is to transition students from proportional models to direct variation models.

rational numbers proportional relationships direct variation models linear functions

BIG IDEAS - NCSM A ratio is a multiplicative comparison of

quantities

Ratios give relative sizes of quantities being compared

Ratios can be expressed as units by finding an equivalent ratio where the second term is one.

BIG IDEAS – NCSM, continued

A proportion is a relationship between relationships.

If two quantities vary proportionally, the ratio of corresponding terms is constant.

If two quantities vary proportionally, the constant ratio can be expressed in lowest terms ( a composite unit) or as a unit amount; the constant ratio is the slope of the related linear function.

Ratio is a reflectively abstracted constant ratio.

LEARNING PATHWAY

REFLECTIONS

In your memo book, describe one thing you will do differently to identify and address students’ misconceptions.

Share an insight that you have identified in the past two days, that will help you clarify misconceptions.

Atlantis Mission One

Student Hat: Complete the activity

and applications in groups of two

Write your answers on the extra

copy.

Use your resources as needed.

Atlantis Mission One

Teacher Hat

What concepts and skills are used in this mission?

What misconceptions might your students have?

How did the applications support student learning?

How does the pre-lesson connect with mission one and the three principles?

Atlantis Mission Two

Teacher Hat

What concepts and skills are used in this mission?

What misconceptions might your students have?

How does the Mission 1 connect with Mission 2 and the three principles?

Mission One & Two Teacher Notes

Take five minutes to review the teacher notes.

Highlight areas you want to remember.

What else would you add to the teacher notes?

Reconnecting with Missions 1 & 2

As a teacher, what connections do you make in regard to mathematical content in the Mission 1&2 activities?

How will these impact your teaching?

Time for Reading

Proportional Reasoning: Student Misconceptions and

Strategies for Teaching

Discussion Questions

1. Why do you think 90% of adults do not reason proportionally, according to Lamon 2007?

2. How are multiplicative situations different from additive situations?

3. What are the key steps to proportional reasoning?

4. What is the difference between ratio & rate?

Implications for Teaching

In what ways will the PCK tool on proportional reasoning impact

your teaching?

As a table group, discuss, chart, and post your ideas on the wall

Atlantis Mission Three

Student Hat: Complete the activity in groups of two

Think about different ways

students might solve the problems.

Write your answers on the extra copy.

Atlantis Mission Three

Teacher HatWhat concepts and skills are used in

this mission?What misconceptions might your

students have?How does Mission Three connect with

the three principles?

Mission Three Teacher Notes

Take five minutes to review the teacher notes.

Highlight areas you want to remember.

What else would you add to the teacher notes?

Reflections

What are the conceptual ideas in

Mission Three?

In your memo book, answer:

How might transitioning from a single lesson lens to a unit (sequence of lessons) learning lens impact teaching and learning in your classroom?

Time for Reading

Proportional Reasoning: Student Misconceptions and

Strategies for TeachingPp. 7- 13

Discussion QuestionsWhat are the common student error

strategies in solving proportional reasoning tasks?

Does the way a problem is presented affect how student interact with the problem? Explain how.

Could additive methods be used in proportional situation?

What are the most common correct strategies in solving proportional tasks?

Mission Four

Student Hat:

Complete the activity in groups of two at your table

Write your answers on the extra copy

Atlantis Mission Four Teacher Hat

How is Mission Four different than the other missions?

What different levels of thinking, cognitive demands, are required to complete Mission Four?

What are the common characteristics of proportional relationships? How do they appear in each representation?

Teacher Notes

Take five minutes to review the teacher notes.

Highlight areas you want to remember.

What else would you add to the teacher notes?

Mission Five

Student Hat:

Complete the activity in groups of two at your table

Write your answers on the extra copy

Atlantis Mission Five

Teacher HatWhat concepts and skills are used in

this mission?What misconceptions might your

students have?How does Mission Five connect with

the three principles?

Mission Six

Student Hat:

Complete the activity in groups of two at your table

Write your answers on the extra copy

Atlantis Mission Six

Teacher HatWhat concepts and skills are used in

this mission?What misconceptions might your

students have?How does Mission Six connect with

the three principles?

What are Key Characteristics

of a Proportional Thinker?

Key Characteristics of a Proportional Thinker

Knowing and understanding the idea of relationship

Ability to recognize multiplicative situations and distinguish it from additive situations

  Ability to recognize and explain the difference

between proportional (y=kx) and non-proportional (y=kx+b) situations

  Ability to recognize and distinguish different types

of proportionality: direct and inverse

Ability to use proportionality as a mathematical model in real-world contexts

Knowing and use of the language of proportionality Understanding the concept of function to express the co-

variation Knowing that the graph of a direct proportional situation is a

straight line that passes through the origin Knowing that the graph of non-proportional situation is a

straight line intersecting the y axis b units from the origin Knowing that the graph of an inversely proportional situation is

a hyperbola Understanding that k is the constant ratio in a direct

proportional situations Understanding that k is the constant product in an inversely

proportional situations

Key Characteristics of a Proportional Thinker (cont.)

Metacognition and Self-Monitoring

Restate Principle Three in your own words. What does it mean to you?

How does the sequence of Mission activities address Principle Three? List at least two examples.

In using metacognition, who is doing the work? What is the teacher doing? What is the student doing?

Post Test

Work on test items individually.Write your participant number on the test.

ReflectionDescribe one significant event, theme or idea from your summer professional development. How is the idea you selected connected to your teaching practice?  How are you planning to implement in your classroom these ideas from your professional development experience for your own benefit and the benefit of your students?

Reflection Questions

Where does this unit fit in my curriculum?

What support will I need to implement this unit?

What will I do differently as a result of my experience in the institute?